Let $W(t_1,\dotsc,t_n)$ a holomorphic function on some connected open set $U$ of $\mathbb C^n$. Let $\mathbf t^{(0)}$ a point of $U$.

Assume that there exists a cycle $\gamma$ in $\mathbb C^m$ and a rational function $F(\mathbf t, x_1,\dotsc, x_m)$ such that for all $\mathbf t$ in a neighbourhood of $\mathbf t^{(0)}$:

- $\mathbf x \mapsto F(\mathbf t, \mathbf x)$ is continuous on $\gamma$;
- $\displaystyle W(\mathbf t) = \oint_\gamma F(\mathbf t, \mathbf x)\mathrm d \mathbf x$.

**Is it true that for all point $\mathbf t^{(1)}$ in $U$, there exists another (sum of) cycle $\gamma_1$ such that properties 1. and 2. are satisfies in a neighbourhood of $\mathbf t^{(1)}$ ?**

For simple integrals, that is $m=1$, this is true, and not too hard to see: the poles of $F(\mathbf t, x)$ are points that move continuously with $\mathbf t$. It is easy to deform $\gamma$ so that is does not encounter these moving points. There is a singularity when a pole inside $\gamma$ collapses with a pole outside, but this can't happen if we stay in the domain of holomorphy of $W$. However, it seems harder for multiple integrals...

Multidimensional Residues and Their Applications(AMS, 1992) books.google.ca/books?id=SKCN1eN7hz0C $\endgroup$